Topological and geometrical methods are known for their capability to reduce noise and have achieved significant success in analyzing complex biological data. A key method in topological data analysis is persistent homology, which leverages a filtration of simplicial complexes to extract multiscale spatial information. To integrate non-spatial information, specially tailored persistent homology approaches, such as element-specific persistent homology, have been proposed and have shown significant success in predictive modeling of molecular structures.Recently, it was discovered that persistent Laplacians can be defined for a filtration, and the nullity of a persistent Laplacian is equal to the corresponding persistent Betti number, suggesting that the spectra of persistent Laplacians offer additional information beyond traditional persistent homology. Spectra of persistent Laplacians can be used in combination with persistent homology to enhance the featurization of raw biological data. Inspired by the theory of cellular sheaves, the theory of persistent sheaf Laplacians was proposed; spectra of persistent sheaf Laplacians encode both spatial and non-spatial information of a labeled point cloud. The theory of persistent sheaf Laplacians provides an elegant method for fusing different types of data and holds significant potential for future development.The construction of persistent Laplacians can also be easily generalized to other settings, such as digraphs and hypergraphs. These generalizations are important, as they offer various ways to integrate different types of biological information. In this thesis, we introduce persistent Laplacians and some generalizations, such as persistent sheaf Laplacians, and discuss their applications in biology.
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